# Does Our Universe Prefer Exotic Smoothness?

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## Abstract

**:**

## 1. Introduction

## 2. Spacetime and Exotic Smoothness

**1.**The cosmos $\sum $ is a compact 3-manifold without boundary.

**2a.**The relative Euler characteristic $\chi (M,\partial M)$ of the spacetime M is zero.

**2b.**The spacetime M is simply connected.

**3.**The cosmos $\sum $ is a homology 3-sphere.

**4.**The spacetime M is a smooth 4-manifold with $\partial M={\sum}_{0}\bigsqcup \sum $, realising a cobordism between two homology 3-spheres.

**Initial state:**- The cosmos begins as a compact 3-manifold ${\sum}_{0}$ without boundary (Condition 1) and possesses the topology of a homology 3-sphere (Condition 2).
**Dynamics:**- The spacetime is a cobordism M with $\partial M={\sum}_{0}\bigsqcup \sum $ (Condition 3). This 4-manifold is simply connected (Condition 2b) and its pseudo-Riemannian metric (Condition 2a) is determined by the Einstein equation. The cosmos expands from ${\sum}_{0}$ to $\sum $ with the scaling factor $a\left(t\right)$ determined by the Friedmann equation. It is interesting to note that cobordisms represent properly spacetime in the categorical approach by John Baez [21]. In Baez’s representation the entire category of spacetime cobordisms (between 3-space manifolds) is considered leading to a natural connection with quantum mechanics (as in topological quantum field theory, TQFT). Even though in our approach the smoothness structures in dimension 4 determine nontrivial cobordisms and we do not discuss the quantum operator representation, still this would be an interesting nontrivial task to find connections with TQFT.
**Topology transition:**- The homology of the cosmos is an invariant (both ${\sum}_{0}$ and $\sum $ are homology 3-spheres, Conditions 2 and 3). The topology of the initial state ${\sum}_{0}$ may change to $\sum $ by a homology-preserving transition (nontriviality of $M\ne \sum \times \mathbb{R}$).

**5.**The initial state ${\sum}_{0}$ is the Einstein cosmos ${S}^{3}$.

**6.**The intermediate state $\sum \left({t}_{1}\right)={S}_{1}$ at ${t}_{0}<{t}_{1}<t$ is the Brieskorn cosmos $\sum (2,5,7)$.

**7.**The K3 surface $\mathcal{K}=K\backslash {D}^{4}$ determines the 4-manifold M with $\partial M={S}^{3}\bigsqcup \sum $ by its common Akbulut cork. M is the physical spacetime.

- $P\#P$ causes the cosmological constant (= dark energy)
- ${K}_{1},{K}_{2},{K}_{3}$ is responsible for the matter part (= three generations?)
- ${S}^{3}$ or ${S}^{2}\times [0,1]$ is associated with the dark matter (in the form of a gravitational soliton?)

## 3. Physical Parameters

**1.**- The $\alpha $ parameter in the Starobinsky model (in the units of the Planck mass squared)$$\alpha \xb7{M}_{P}^{-2}=\frac{1}{\left(1+\vartheta +\frac{{\vartheta}^{2}}{2}+\frac{{\vartheta}^{3}}{6}\right)}\approx {10}^{-5}\phantom{\rule{4.pt}{0ex}}\mathrm{where}\phantom{\rule{4.pt}{0ex}}\vartheta =\frac{3}{2\xb7CS(\sum (2,5,7\left)\right)}=\frac{140}{3}\phantom{\rule{0.166667em}{0ex}}.$$
**2.**- The number of e-folds during the inflation$$N=\frac{3}{2\xb7CS(\sum (2,5,7\left)\right)}+\mathrm{ln}\left(8\pi \right)\approx 51\phantom{\rule{0.166667em}{0ex}}.$$
**3.**- The scalar/tensor ratio $r=\frac{12}{{(\vartheta +ln\left(8{\pi}^{2}\right))}^{2}}\approx 0.0046\phantom{\rule{0.277778em}{0ex}}.$
**4.**- The spectral tilt ${n}_{s}=1-\frac{2}{\vartheta +ln\left(8{\pi}^{2}\right)}\approx 0.961\phantom{\rule{0.277778em}{0ex}}.$
**5.**- The GUT energy scale (the energy of the first topology change ${S}^{3}\to \sum (2,5,7))$$$\Delta {E}_{1}=\frac{{E}_{\mathrm{Plank}}}{1+\vartheta +\frac{{\vartheta}^{2}}{2}+\frac{{\vartheta}^{3}}{6}}\approx {10}^{15}\phantom{\rule{0.166667em}{0ex}}\mathrm{Ge}\mathrm{V}\phantom{\rule{0.166667em}{0ex}}.$$
**6.**- The electroweak energy scale (the energy assigned to the second topological transition $\sum (2,5,7)\to P\#P$)$${E}_{2}=\frac{{E}_{\mathrm{Plank}}\xb7\mathrm{exp}\left(-\frac{1}{2\xb7CS(P\#P)}\right)}{1+\vartheta +\frac{{\vartheta}^{2}}{2}+\frac{{\vartheta}^{3}}{6}}\approx 63\phantom{\rule{0.166667em}{0ex}}\mathrm{Ge}\mathrm{V}\phantom{\rule{0.277778em}{0ex}}.$$
**7.**- The topological bound on the sum of the three neutrino masses $<0.018\phantom{\rule{0.166667em}{0ex}}\mathrm{e}\mathrm{V}\phantom{\rule{0.166667em}{0ex}}.$

Let W be a simply connected compact manifold with a boundary $\partial M$ that has two components, ${M}_{1}$ and ${M}_{2}$ such that the inclusions ${i}_{1,2}:{M}_{1,2}\hookrightarrow M$ are homotopy equivalences. Then W is diffeomorphic to the product ${M}_{1}\times [0,1]={M}_{2}\times [0,1]$, where dimensions of ${M}_{1,2}\ge 5$. This means that if ${M}_{1}$ and ${M}_{2}$ are two simply connected manifolds of dimension $\ge 5$ and there exists an h-cobordism W between them, then W is a product ${M}_{1}\times [0,1]$ and ${M}_{1}$ is diffeomorphic to ${M}_{2}$.

There exist simply connected compact cobordisms W of dimension 5 with the inclusions of their boundary components ${M}_{1,2}\stackrel{{i}_{1,2}}{\hookrightarrow}W$ being homotopy equivalences such that W is not diffeomorphic to the product ${M}_{1}\times [0,1]$ (or ${M}_{2}\times [0,1]$) hence ${M}_{1}$ is not diffeomorphic to ${M}_{2}$ being h-cobordant to it.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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Asselmeyer-Maluga, T.; Król, J.; Miller, T.
Does Our Universe Prefer Exotic Smoothness? *Symmetry* **2020**, *12*, 98.
https://doi.org/10.3390/sym12010098

**AMA Style**

Asselmeyer-Maluga T, Król J, Miller T.
Does Our Universe Prefer Exotic Smoothness? *Symmetry*. 2020; 12(1):98.
https://doi.org/10.3390/sym12010098

**Chicago/Turabian Style**

Asselmeyer-Maluga, Torsten, Jerzy Król, and Tomasz Miller.
2020. "Does Our Universe Prefer Exotic Smoothness?" *Symmetry* 12, no. 1: 98.
https://doi.org/10.3390/sym12010098